Aspects of the (0,2)-McKay Correspondence

Abstract

We study first order deformations of the tangent sheaf of resolutions of Calabi-Yau threefolds that are of the form $\CC^3/\ZZ_r$, focusing on the cases where the orbifold has an isolated singularity. We prove a lower bound on the number of deformations of the tangent bundle for any crepant resolution of this orbifold. We show that this lower bound is achieved when the resolution used is the G-Hilbert scheme, and note that this lower bound can be found using a combinatorial count of (0,2)-deformation moduli fields forN=(2,2) conformal field theories on the orbifold. We also find that in general this minimum is not achieved, and expect the discrepancy to be explained by worldsheet instanton corrections coming from rational curves in the orbifold resolution. We show that irreducible toric rational curves will account for some of the discrepancy, but also prove that there must be additionalworldsheet instanton corrections beyond those from smooth isolated rational curves.